Set Math 9

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Specification Grid Grade 9-10

S.N.	Areas	Working hours	Knowledge		Understanding		Application		Higher ability		Total items	questions	Marks
			Items	Marks	Items	Marks	Items	Marks	Items	Marks
1.	Sets	12	1	1	1	1	1	3	1	1	4	1	6

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कक्षागत सिकाइ उपलब्धि ९ र १०

क्र.स.	विषयवस्तुको क्षेत्र	कक्षा ९	कक्षा १०
१.	समूह	समूहका क्रियाहरु गर्न र भेन चित्रमा प्रस्तुत गर्न समूहको गणनात्मकता पत्ता लगाउन	समूहका क्रियाहरू, भेनचित्र र गणनात्मकताको प्रयोग गरी तीनओटासम्म समूहसँग सम्बन्धित व्यावहारिक समस्याहरू समाधान गर्न

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Scope and Sequence of Contents of Grade 9

समूह

1. समूहका क्रियाहरू संयोजन, प्रतिच्छेदन, फरक र पुरक (तीनओटासम्म समूह)
2. समूहको गणनात्मकता

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Table of Contents

1. Set and Notation
1. Describing a Set
2. Relation Between Sets
2. Set Operation
1. Union of Sets
2. Intersection of Sets
3. Difference of Sets
4. Complement of Sets
3. Parts of the sets
4. Cardinality of Set
1. Arithemetic of Cardinality
2. Test your Understandings

Set and Notation

Set theory is a branch of mathematics. It studies the properties of collection of well-defined objects.
A German mathematician Georg Cantor (1845–1918) has conceptualized the modern study of set theory. According to him set is a collection of well-defined objects in which it is possible to determine if a given object is included in the collection.
We use \( \in \) and \( \notin \) symbols to represent if an element is included to a set or not respectively.
For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and C = {n² − 4 : n is an integer; and 0 ≤ n ≤ 19} , we can write
4 ∈ A and 285 ∈ C; but
9 ∉ C and green ∉ B.

Set is denoted by a single capital letter (upper cases) of English alphabets such as A, B, C … and so forth. For example
A = {a, e, i, o, u} :The set of vowels

The objects in a set are known as elements or members of the set. The elements in a set are enclosed within middle brackets. For instance, a set A of vowels in English is written within {…} and it is written as
A = {a, e, i, o, u}.

The elements of set are denoted by small letters of English alphabets unless and otherwise stated if applicable. However, the elements in a set can be material objects such as books, pens, people etc. or conceptual objects such as numbers,alphabets, points etc.
Please lease be careful to distinguish the symbols \( 0, \{0\}, \phi, \{\phi\} \)

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Describing a Set

When we specify elements of a set, we are simply describing the set. The most common methods used to describe sets are given below.

1. Semantic definition/ Intentional definition/ Verbal description

   It is also called intentional definition, using a rule or semantic description. This is called verbal description method. For Example

   1. A is the set of first four positive integers.
   2. B is the set of colors used in Nepali flag.

2. Syntatic definition/ Extensional definition / Listing method

   This method is called roster notation or listing method done by listing each member of the set. This extensional definition is denoted by enclosing the list of members in curly brackets. For Example

   1. C = {4, 2, 1, 3}
   2. D = {blue, white, red}

   In this method, every element of a set must be unique; no two members may be identical. However, the order in which the elements of a set are listed is ignored (unlike for a sequence or tuple).
   Combining these two ideas into an example
   A={6, 11} , B= {11, 6}
   In the examples above, A = B.

3. Venn-Diagram

   In this method, a diagram is used to represent the relationship among the sets, called Venn-diagram. It was named after an English philosopher John Venn (1834-1923). In a Venn-Diagram, rectangular region represent universal set, and other subsets usually by circular regions (sometimes, it can be expressed by ellipse, sphere etc). For Example
   If U={1,2,3,4,5,6,7,8,9,10}
   A={1,2,3,4,5}
   B={4,5,6,7,8}
   Then, the Venn-Diagram can be described as below.

4. The set-builder notation

   1. A={x:x is the set of first four positive integers}
   2. B={x|x is the set of colors used in Nepali flag}

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Relation Between Sets

In mathematics, a relation between sets is a subset of their Cartesian product. However, we can also define relation as follows.

1. Subset

   A set \( A\) is said to be a subset of another set \( B\) if every element of \( A\) is also an element of the set \( B\) . If \( A\) is subset of \( B\) , then it is written as \(A \subset B\) and read as “\( A\) is contained in \( B\) ” or “\( A\) is a subset of \( B\) ”.
   If set A is NOT subset of set B, then it is written as \(A \not\subset B\).
   For example:
   Let \(A = \{1, 2, 3\}, B = \{3, 4, 5,6\}\) and \(C = \{1, 2, 3, 4, 5\}\) then \(A \subset C\) but \(B \not\subset C\).
   In usual notation of set of numbers, the relation between them are as below
   \(\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset\mathbb{Q} \subset \mathbb{R} \).
   There are two types of subsets

   1. Proper subset
      If \( A \subset B\) and \( A \ne B\) then \( A\) is called a proper subset of \( B\) . In this case, \( B\) is superset of \( A\)
   2. Improper subset
      If \( A \subset B\) and \( A = B\) then \( A\) is called an improper subset of \( B\) . It is written as \( A \subseteq B\) .
      Note:
   • The empty set \( \phi \) is a subset of every set.
   • Every set is a subset of itself.
   • Every non-empty set has at least two subsets
   • The total number of possible subsets of a set with n-elements is \( 2^n\) .

2. Power Set

   Let \( S\) is a set. Then the set of all the possible subsets of \( S\) is called power set of \( S\) . It is denoted by \( P(S)\) . For example,
   if \( S = \{a, b, c\}\) then \( P(S) = \{\phi, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\},\{a, c\}, \{a, b, c\}\}\) .

   Thus, the power set of a set S is the set of all subsets of S, including S itself and the empty set. For example, the power set of the set {1, 2, 3} is
   {{1, 2,3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}.
   The power set of a set S usually written as P(S).

   The power set of a finite set with n elements has 2ⁿ elements.
   This relationship is one of the reasons for the terminology power set. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains
   2³ = 8 elements.
   The power set of an infinite (either countable or uncountable) set is always uncountable.

   • If \( P (S)\) is the power set of a set \( S\) then \( n (P(S)) = 2^{n(S)}\) .
   • Power set of a finite set is finite.
   • \( S \in P(S)\) , that is, Set \( S\) is an element of power set of \( S\) .

3. Equal Sets

   Two or more sets are called equal (or identical or same) if they consist same elements. For example,
   if \( A = \{1, 2\}\) and \( B = \{2, 1\}\) then \( A = B\) but the sets \( C = \{12\}\) and \( D =\{21\}\) are not equal.

4. Equivalent Sets

   Two sets \( A\) and \( B\) are called equivalent if their cardinal number is same, i.e., \( n (A) = n (B)\) . The symbol to denote equivalent sets is “\( \sim\) ”. For example,
   if \( A = \{1, 2, 3\}\) and \( B = \{p, q, r\}\) then \( A \sim B\).

5. Overlapping Sets

   Two sets \( A\) and \( B\) are called overlapping set if they do have some common element. For example,
   if \( A = \{1,2,3\}\) and \( B = \{3,4,5\}\) then \( A\) and \( B\) are overlapping sets as \( \{3\}\) is common to both sets \( A\) and \( B\) .

6. Disjoint Sets

   Two sets \( A\) and \( B\) are called disjoint if they have no elements in common. For example, if \( A = \{1, 2, 3\}\) and \( B = \{4,5, 6\}\) then \( A\) and \( B\) are disjoint as they have no element in common.

7. Comparable Sets

   Two sets \( A\) and \( B\) are said to be comparable if \( A \subset B \) or \( B \subset A\) . For example, the sets \( A=\{a, b, c\}\) , and \( C=\{a, b, c, d\}\) are comparable. But the sets \( C=\{a, b, c\}\) and \( D=\{a, c, d, e\}\) are not comparable (incomparable) sets because C and D are overlapping sets.

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Set Operation

In real number system, we can do four fundamental operation to form new number by combining or manipulating one or more existing numbers. For example, given two numbers \(2\) and \(3\) , we can use

1. \(+\) to form a new number \(5\) by \(2+3\)
2. \(\times\) to form a new number \(6\) by \(2 \times 3\)
3. we can do Set operation to form new Set by combining or manipulating one or more existing Sets.
4. Set operation helps to combine two or more sets together to form a new set.
5. The common example of set operations are: Union, Intersection, Difference, and Complement

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Union of Two Sets

Let A and B be any two sets. Then union of sets A and B is a new set consisting all the elements of A and B without repetition. The union is the smallest set containing elements of A and B.
In other words
The union of two sets A and B is the set of elements which are in A, in B, or in both A and B
It is denoted by AUB and read as “A union B” or “A cup B”.
Mathematically,
AUB = {x: x ∈ A or x ∈ B}.

मानौ A र B कुनै दुई समुहहरू छन । अब समुह A र B को संयोजन (union) भनेको एउटा नयाँ समुह हो जुन A र B का सबै सदस्यहरु समावेश भई बनेको हुन्छ। संयोजन समुह A र B बाट बन्ने सबैभन्दा सानो समुह हो । यसलाई AUB ले जनाईन्छ र "A संयोजन B" भनेर पढिन्छ।

गणितिय भाषामा,
AUB = {x: x ∈ A or x ∈ B}.

1. Example 1

   If A={ 1,2,3,4,5} and B={4,5,6,7,8}, then find A∪B
   Solution
   In this example, A={ 1,2,3,4,5} and B={4,5,6,7,8}
   Thus,
   A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
   or A∪B={4,5} ∪{1,2,3}∪{6,7,8}
   or A∪B={1,2,3,4,5,6,7,8}

   A∪B by shaded region

2. Example 2

   If A={ 1,2,3} and B={6,7,8}, then find A∪B
   Solution
   In this example, A={ 1,2,3} and B={6,7,8}
   Thus,
   A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
   or A∪B={ }∪{1,2,3}∪{6,7,8}
   or A∪B={1,2,3,6,7,8}

   the shaded region is A∪B

3. Example 3

   If A={ 1,2,3,4,5} and B={4,5}, then find A∪B
   Solution
   In this example, A={1,2,3,4,5} and B={4,5}
   Thus,
   A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
   or A∪B={4,5} ∪{1,2,3}∪{}
   or A∪B={1,2,3,4,5}

   the shaded region is A∪B

4. Example 4

   If B={ 1,2,3,4,5} and A={4,5}, then find A∪B
   Solution
   In this example, B={1,2,3,4,5} and A={4,5}
   Thus,
   A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
   or A∪B={4,5} ∪{1,2,3}∪{}
   or A∪B={1,2,3,4,5}

   the shaded region is A∪B

5. Example 5

   If A={1,2,3,4,5} and B={1,2,3,4,5}, then find A∪B
   Solution
   In this example, A={1,2,3,4,5} and B={1,2,3,4,5}
   Thus,
   A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
   or A∪B={1,2,3,4,5} ∪{}∪{}
   or A∪B={1,2,3,4,5}

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Union of Three Sets

Let A, B and C be any three sets. Then union of sets A, B and C is a new set consisting all the elements of A, B and C without repetition. The union is the smallest set containing elements of A, B and C.
In other words
The union of three sets A, B and C is the set of elements which are in A, in B, in C or in both A, B and C
It is denoted by AUBUC and read as “A union B union C” or “A cup B cup C”.
Mathematically,
AUBUC = {x: x ∈ A or x ∈ B or x ∈ C}.

मानौ A, B र C कुनै तिन समुहहरू छन । अब समुह A, B र C को संयोजन (union) भनेको एउटा नयाँ समुह हो जुन A, B र C का सबै सदस्यहरु समावेश भई बनेको हुन्छ। संयोजन समुह A, B र C बाट बन्ने सबैभन्दा सानो समुह हो । यसलाई AUBUC ले जनाईन्छ र "A संयोजन B संयोजन C " भनेर पढिन्छ।

गणितिय भाषामा,
AUBUC = {x: x ∈ A or x ∈ B or x ∈ C}.
समूहको संयोजन गर्दा दिइएका समूहका साझा सदस्यहरूलाई नदोहो-याइकन बाँकी सबै सदस्यहरूलाई लिएर समूहको रूपमा लेख्नुपर्छ ।

Example 1

If U={a, b, c, d, e,f,g,h,i,o,u}, A = {a, b, c, d, e}, B = {a, e, i, o, u}, C = {d, e, f, g} are given then find \(A \cup B \cup C\) and present it in Venn-Diagram.
Given that
U={a, b, c, d, e,f,g,h,i,o,u}
A = {a, b, c, d, e}
B = {a, e, i, o, u}
C = {d, e, f, g}
The union of A,B and C is given by
AUBUC = {x: x ∈ A or x ∈ B or x ∈ C}.
or AUBUC = {a, b, c, d, e,f,g,i,o,u}
सँगैको भेनचित्रमा छाया पारेको भागले AUBUC लाई जनाउँछ ।

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Intersection of Sets

Let A and B be any two sets. Then intersection of sets A and B is a new set consisting common elements of A and B. The intersection is the largest set containing common elements of A and B.
It is denoted by A∩B and read as “A intersection B” or “A cap B”.
Mathematically, A∩B = {x: x ∈ A and x ∈ B}.

मानौ A र B कुनै दुई समुहहरू छन । अब समुह A र B को प्रतिच्छेदन (intersection) भनेको एउटा नयाँ समुह हो जुन A र B का सबै साझा सदस्यहरु समावेश भई बनेको हुन्छ। प्रतिच्छेदन समुह A र B को साझा सदस्यबाट बन्ने सबैभन्दा ठुलो समुह हो । यसलाई A∩B ले जनाईन्छ र "A प्रतिच्छेदन B" भनेर पढिन्छ।

गणितिय भाषामा, A∩B = {x: x ∈A and x ∈ B}.

1. Example 1

   If A={ 1,2,3,4,5} and B={4,5,6,7,8}, then find A∩B
   Solution
   In this example, A={ 1,2,3,4,5} and B={4,5,6,7,8}
   Thus,
   A∩B ={Common Elements of A and B}
   or A∩B ={4,5}
   or A∩B ={4,5}

   the shaded region is A∩B

2. Example 2

   If A={ 1,2,3} and B={6,7,8}, then find A∩B
   Solution
   In this example, A={ 1,2,3} and B={6,7,8}
   Thus,
   A∩B ={Common Elements of A and B}
   or A∩B ={ }
   or A∩B ={ }

   the shaded region is A∩B , Empty Set

3. Example 3

   If A={ 1,2,3,4,5} and B={4,5}, then find A∩B
   Solution
   In this example, A={1,2,3,4,5} and B={4,5}
   Thus,
   A∩B ={Common Elements of A and B}
   or A∩B ={4,5}
   or A∩B ={4,5}

   the shaded region is A∩B

4. Example 4

   If B={ 1,2,3,4,5} and A={4,5}, then find A∩B
   Solution
   In this example, B={1,2,3,4,5} and A={4,5}
   Thus,
   A∩B ={Common Elements of A and B}
   or A∩B ={4,5}
   or A∩B ={4,5}

   the shaded region is A∩B

5. Example 5

   If A={1,2,3,4,5} and B={1,2,3,4,5}, then find A∩B
   Solution
   In this example, A={1,2,3,4,5} and B={1,2,3,4,5}
   Thus,
   A∩B ={Common Elements of A and B}
   or A∩B ={1,2,3,4,5}
   or A∩B ={1,2,3,4,5}

   the shaded region is A∩B

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Intersection of Three Sets

Let A, B and C be any three sets. Then intersection of sets A, B and C is a new set consisting all the COMMON elements of A, B and C without repetition. The union is the largest set containing the COMMON elements of A, B and C.
In other words
The intersection of three sets A, B and C is the set of elements which are in A, and in B, and in C
It is denoted by A∩B∩C and read as “A intersection B intersection C” or “A cap B cap C”.
Mathematically,
A∩B∩C = {x: x ∈ A and x ∈ B and x ∈ C}.

मानौ A, B र C कुनै तिन समुहहरू छन । अब समुह A, B र C को प्रतिच्छेदन (intersection) भनेको एउटा नयाँ समुह हो जुन A, B र C का सबै साझा सदस्यहरु समावेश भई बनेको हुन्छ। प्रतिच्छेदन समुह A, B र C को साझा सदस्यबाट बन्ने सबैभन्दा ठुलो समुह हो । यसलाई A∩B∩C ले जनाईन्छ र "A प्रतिच्छेदन B प्रतिच्छेदन C " भनेर पढिन्छ।

गणितिय भाषामा,
A∩B∩C = {x: x ∈ A and x ∈ B and x ∈ C}.
समूहको प्रतिच्छेदन गर्दा दिइएका सबै समूहका साझा सदस्यहरूलाई मात्र नदोहो-याइकन समूहको रूपमा लेख्नुपर्छ ।

Example 1

If U={a, b, c, d, e,f,g,h,i,o,u}, A = {a, b, c, d, e}, B = {a, e, i, o, u}, C = {d, e, f, g} are given then find \(A \cap B \cap C\) and present it in Venn-Diagram.
Given that
U={a, b, c, d, e,f,g,h,i,o,u}
A = {a, b, c, d, e}
B = {a, e, i, o, u}
C = {d, e, f, g}
The union of A,B and C is given by
A∩B∩C = {x: x ∈ A and x ∈ B and x ∈ C}.
or A∩B∩C = {a, b, c, d, e}∩{a, e, i, o, u}∩{d, e, f, g}
or A∩B∩C = {e}
सँगैको भेनचित्रमा घेरा पारेको भागले A∩B∩C लाई जनाउँछ ।

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Difference of Sets

Let A and B be any two sets. Then difference of sets A and B is a new set consisting elements of only A which are NOT in B.
It is denoted by A-B and read as “A difference B” or “A - B”.
Mathematically, A-B = {x: x ∈ A and x ∉ B}.

मानौ A र B कुनै दुई समुहहरू छन । अब समुह A र B को फरक (difference) भनेको एउटा नयाँ समुह हो जुन A मा मात्र भएको तर B मा नभएको सबै सदस्यहरु समावेश भई बनेको हुन्छ। यसलाई A-B ले जनाईन्छ र "A फरक B" भनेर पढिन्छ।

गणितिय भाषामा, A-B = {x: x ∈A and x ∉ B}.
The union of A-B and B-A is called symmetric difference of A and B, and it is denoted by \(A \triangle B\) or \(A \ominus B\), and read as " A symmetric difference B".

1. Example 1

   If A={ 1,2,3,4,5} and B={4,5,6,7,8}, then find A-B
   Solution
   In this example, A={ 1,2,3,4,5} and B={4,5,6,7,8}
   Thus,
   A-B =Elements in Abut NOT in B
   or A-B ={1,2,3}
   or A-B ={1,2,3}

2. Example 2

   If A={ 1,2,3} and B={6,7,8}, then find A-B
   Solution
   In this example, A={ 1,2,3} and B={6,7,8}
   Thus,
   A-B =Elements in Abut NOT in B
   or A-B ={1,2,3}
   or A-B ={1,2,3}

3. Example 3

   If A={1,2,3,4,5} and B={4,5}, then find A-B
   Solution
   In this example, A={1,2,3,4,5} and B={4,5}
   Thus,
   A-B =Elements in Abut NOT in B
   or A-B ={1,2,3}
   or A-B ={1,2,3}

4. Example 4

   If B={1,2,3,4,5} and A={4,5}, then find A-B
   Solution
   In this example, B={1,2,3,4,5} and A={4,5}
   Thus,
   A-B =Elements in Abut NOT in B
   or A-B ={}
   or A-B ={}

5. Example 5

   If A={1,2,3,4,5} and B={1,2,3,4,5}, then find A-B
   Solution
   In this example, A={1,2,3,4,5} and B={1,2,3,4,5}
   Thus,
   A-B =Elements in Abut NOT in B
   or A-B ={}
   or A-B ={}

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Complement of Set

Let A and B be any two sets. Then Complement of sets A is a new set consisting elements which are NOT in A.
It is denoted by A' or \(\overline{A}\) and read as “A Complement” or “U - A”.
Mathematically, A' = {x: x ∈ U and x ∉ A}.

मानौ A कुनै एउटा समुह हो । अब समुह A को पुरक (Complement) भनेको एउटा नयाँ समुह हो जुन A मा नभएको सबै सदस्यहरु समावेश भई बनेको हुन्छ। यसलाई A' or \(\overline{A}\) ले जनाईन्छ र "U-A" भनेर पढिन्छ।

गणितिय भाषामा, A' = {x: x ∈ U and x ∉ A}.

1. Example 1

   If U={ 1,2,3,4,5,6,7,8,9,10} with A={1,2,3,4,5}, B={4,5,6,7,8} , then find A'
   Solution
   In this example,
   A'= U-A={6,7,8,9,10}

2. Example 1

   If U={ 1,2,3,4,5} and A={4,5} , then find A'
   Solution
   In this example,
   U={ 1,2,3,4,5} and A={4,5}
   Therefore, A'= U-A={1,2,3}

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Parts of the Two sets

Below is a Venn diagram involving two sets A and B

We can make four disjoint parts of the above Venn-diagram, which are as below.

1. Part 1: A-B

   This part is known as A difference B
   It is denoted by A−B
   It is also denoted by A_o
   It represents the cardinality (or elements) which lies in only A but not in B

2. Part 2: A∩B

   This part is also known as A intersection B
   This part is denoted by A∩B
   This parts represents the cardinality (or elements) which lies both in A and B.

3. Part 3: B-A

   This part is also known as B difference A
   This part is denoted by B-A
   This part is also denoted by B_o
   This parts represents the cardinality (or elements) which lies in only B but not in A

4. Part 4: (AUB)'

   This part is also known complement of A union B
   This parts represents the cardinality (or elements) which lies neither in A nor in B

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Different notion of sets using four disjoint parts

Below is a Venn diagram involving two sets A and B

Here are four disjoint parts of the Venn-diagram. These four parts are

• A₀ (or A-B)[red color]
• B₀ (or B-A)[green Color]
• A∩B [gray Color]
• (AUB)' [yellow color]

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Now,how many different sets can be formed using these four disjoint parts.
Using these four disjoint parts, all together 16 different set notations can be formed. More explicitly

1. 1 set notation can be formed taking 0 parts out of 4 disjoint parts
2. four different set notations can be formed taking 1 parts out of 4 disjoint parts
3. six different set notations can be formed taking 2 parts out of 4 disjoint parts
4. four different set notations can be formed taking 3 parts out of 4 disjoint parts
5. 1 set notation can be formed taking 4 parts out of 4 disjoint parts

Possible Combinations	Possible number of sets	Set Notations
Set with zero parts	1	\( \phi \)
Set with one parts	4	\(A_0,B_0,A \cap B, (A \cup B)' \)
Set with two parts	6	\( A,B,A',B,A \triangle B, (A \triangle B)'\)
Set with three parts	4	\( (A-B)', (B-A)',(A \cap B)', A \cup B \)
Set with four parts	1	\( U \)

अब, माथिको भेन चित्रको आधारमा चारवटा अलगिएका समुहहरुलाई प्रयोग गरेर कतिवटा फरक फरक समुहहरु बनाउन सकिन्छ?
माथिको चारवटा अलगिएका समुहहरुलाई प्रयोग गरेर जम्मा 16 वटा फरक फरक समुहहरु बनाउन सकिन्छ । जसमा

1. 0 वटा भागलाई प्रयोग गरेर १ वटा समुह बनाउन सकिन्छ।
2. १ वटा भागलाई प्रयोग गरेर ४ वटा समुह बनाउन सकिन्छ।
3. २ वटा भागलाई प्रयोग गरेर ६ वटा समुह बनाउन सकिन्छ।
4. ३ वटा भागलाई प्रयोग गरेर ४ वटा समुह बनाउन सकिन्छ।
5. ४ वटा भागलाई प्रयोग गरेर १ वटा समुह बनाउन सकिन्छ।

Therefore, all together 16 different set notation can be formed.
These 16 different set notation are given below.

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1. Part 1: 𝜙

   Set Notation:𝜙
   This part is formed taking 0 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as empty set.
   It contains no cardinality (or elements) of the sets A or B or U.
   समुह संकेत :𝜙
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये 0 वटा भाग को प्रयोग भएको छ।
   यसलाई खाली समुह पनि भनिन्छ ।
   यस समुहमा समुह A वा B वा U कुनैमा पनि नभएका सदस्यहरु पर्दछन।

2. Part 2: A-B

   Set Notation:A-B (or A₀)
   This part is formed taking 1 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as A difference B.
   It contains the cardinality (or elements) that belong to set A only, but not in B
   समुह संकेत :A-B (or A₀)
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
   यसलाई A र B को फरक समुह भनिन्छ ।
   यसमा समुह A मा भएका तर B मा नभएका सदस्यहरु पर्दछन।

3. Part 3: B-A

   Set Notation:B-A (or B₀)
   This part is formed taking 1 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as B difference A.
   It contains the cardinality (or elements) that belong to set B only, but not in A
   समुह संकेत :B-A (or B₀)
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
   यसलाई B र A को फरक समुह भनिन्छ ।
   यसमा समुह B मा भएका तर A मा नभएका सदस्यहरु पर्दछन।

4. Part 4: A∩B

   Set Notation:A∩B
   This part is formed taking 1 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as A intersection B.
   It contains the cardinality (or elements) that belong to sets A and B both
   समुह संकेत :A∩B
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
   यसलाई A र B को प्रतिच्छेदन समुह भनिन्छ ।
   यसमा समुह A र B दुवैमा पर्ने साझा सदस्यहरु पर्दछन।

5. Part 5: (AUB)'

   Set Notation: (AUB)'
   This part is formed taking 1 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as complement of A union B.
   It contains the cardinality (or elements) that belong to sets Neither A nor B nor both
   समुह संकेत : (AUB)'
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
   यसलाई A र B को संयोजन को पुरक समुह भनिन्छ ।
   यसमा समुह A वा B दुवैमा नपर्ने सदस्यहरु पर्दछन।

6. Part 6: A

   Set Notation: A
   This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as A .
   It contains the cardinality (or elements) that belong to sets A
   समुह संकेत :A
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
   यसलाई A समुह भनिन्छ ।
   यसमा समुह Aमा पर्ने सबै सदस्यहरु पर्दछन।

7. Part 7: B

   Set Notation:B
   This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as B .
   It contains the cardinality (or elements) that belong to sets B
   समुह संकेत :B
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
   यसलाई B समुह भनिन्छ ।
   यसमा समुह B मा पर्ने सबै सदस्यहरु पर्दछन।

8. Part 8: A'

   Set Notation:A'
   This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as complement of A .
   It contains the cardinality (or elements) that does NOT belong to set A
   समुह संकेत :A'
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
   यसलाई A को पुरक समुह भनिन्छ ।
   यसमा समुह A मा नपर्ने सबै सदस्यहरु पर्दछन।

9. Part 9: B'

   Set Notation:B'
   This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
   This part is also known as complement of B .
   It contains the cardinality (or elements) that does NOT belong to set B
   समुह संकेत :B'
   यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
   यसलाई B को पुरक समुह भनिन्छ ।
   यसमा समुह B मा नपर्ने सबै सदस्यहरु पर्दछन।

10. Part 10: A∆B

    Set Notation:A∆B
    This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as union of (A-B) and (B-A) .
    It contains the cardinality (or elements) that belong to only one set
    समुह संकेत :A∆B
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई (A-B) र (B-A) को संयोजन समुह भनिन्छ ।
    यसमा समुह (A-B) वा (B-A) मा पर्ने सबै सदस्यहरु पर्दछन।

11. Part 11: (A∆B)'

    Set Notation: (A∆B)'
    This part is formed taking 2 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as complement of the union of (A-B) and (B-A) .
    It contains the cardinality (or elements) that does NOT belong to only one set
    समुह संकेत : (A∆B)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई (A-B) र (B-A) को संयोजनको पुरक समुह भनिन्छ ।
    यसमा समुह (A-B) र (B-A) कुनैमा पनि नपर्ने सदस्यहरु पर्दछन।

12. Part 12: AUB

    Set Notation: AUB
    This part is formed taking 3 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as union of A and B.
    It contains the cardinality (or elements) that belongs to either A or B or Both
    समुह संकेत : AUB
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को संयोजन समुह भनिन्छ ।
    यसमा समुह A वा B मा पर्ने सदस्यहरु पर्दछन।

13. Part 13: \(A_0'\)

    Set Notation: (A₀)'
    This part is formed taking 3 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as complement of A difference B.
    It contains the cardinality (or elements) that does Not belong to A only
    समुह संकेत : (A₀)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को फरक को पुरक समुह भनिन्छ ।
    यसमा समुह A र B को फरकमा नपर्ने सबै सदस्यहरु पर्दछन।

14. Part 14: \(B_0'\)

    Set Notation: (B₀)'
    This part is formed taking 3 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as complement of B difference A.
    It contains the cardinality (or elements) that does Not belong to B only
    समुह संकेत : (B₀)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई B र A को फरक को पुरक समुह भनिन्छ ।
    यसमा समुह B र A को फरकमा नपर्ने सबै सदस्यहरु पर्दछन।

15. Part 15: (A∩B)'

    Set Notation: (A∩B)'
    This part is formed taking 3 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as complement of A intersection B.
    It contains the cardinality (or elements) that does Not belong to both A and B
    समुह संकेत : (A∩B)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई B र A को प्रतिच्छेदन को पुरक समुह भनिन्छ ।
    यसमा समुह A र B को प्रतिच्छेदनमा नपर्ने सबै सदस्यहरु पर्दछन।

16. Part 16: U

    Set Notation:U
    This part is formed taking 4 parts out of the four parts A₀, B₀,A∩B and (AUB)'
    This part is also known as full (Universal) set.
    It contains all cardinality (or elements) of the sets A or B or U.
    समुह संकेत :U
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A₀, B₀,A∩B र (AUB)'मध्ये ४ वटा भाग को प्रयोग भएको छ।
    यसलाई सर्वव्यापक समुह भनिन्छ ।
    यसमा समुह A वा B वा U भएका सबै सदस्यहरु पर्दछन।

Different Set notations involving Three Sets

मानौ, सर्वव्यापक समुह U को उपसमुहहरु A,B र C छन भने तिन वटा समुहहरु समावेस भएका समस्याहरु समाधान गर्न तलको भेन चित्र प्रयोग गर्नुहोस। (Let A, B and C are the subsets of an universal set U, then use the following Venn-diagram to solve problems related to three sets.

1. \( n_o(A)=p\)
   \(n(A-B-C)=p\)
   This part is also known as A difference with B and C as denoted by A-B-C. This parts represents the cardinality (or elements) which lies in only in A but niether in B nor in C.
2. \( n_o(B)=q\)
   \( n(B-C-A)=q\)
   This part is also known as B difference with C and A as denoted by B-C-A. This parts represents the cardinality (or elements) which lies in only in B but niether in C nor in A.
3. \( n_o(C)=r\)
   \(n(C-A-B)=r\)
   This part is also known as C difference with A and B as denoted by C-A-B. This parts represents the cardinality (or elements) which lies in only in C but niether in B nor in A.
4. \( n_o(A \cap B)=s\)
   \(n((A \cap B)-C)=s\)
   This part is also known as intersection of A and B, only. This parts represents the cardinality (or elements) which lies in only intersection of A and B but NOt in C.
5. \( n_o(B \cap C)=t\)
   \( n((B \cap C)-A)=t\)
   This part is also known as intersection of B and C, only. This parts represents the cardinality (or elements) which lies in only intersection of B and C but NOT in A.
6. \( n_o(A \cap C)=u\)
   \( n(A \cap C)-B)=u\)
   This part is also known as intersection of A and C, only. This parts represents the cardinality (or elements) which lies in only intersection of A and C but NOT in B.
7. \(n(A \cap B \cap C)=v\)
   This part is also known as intersection of A , B and C. This parts represents the cardinality (or elements) which lies in A, B and C, in all three sets.
8. \(\overline{AUBUC}=w\)
   This part is also known as complement of union of A , B and C. It is also denoted by \( (A \cup B \cup C)'\) or \( (A \cup B \cup C)^c\). This parts represents the cardinality (or elements) which does NOT lier on either A or B or C.

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Cardinality of Set

The concept and notation of Cardinality are due to Georg Cantor who defined the notion of cardinality and realized that sets can have different cardinalities. In summary,

1. The cardinality of finite set A is \(n(A)\)
2. The cardinality of countable set is \(\aleph_0\) (read as aleph-naught or aleph-zero or aleph-null)
3. The cardinality of uncountable set is \(𝑐\) (read as continuum)

The cardinality of a set A is the number of elements of the set A . The cardinality of a set A is usually denoted by n(A) but it can also be denoted as Card(A). For example:

• If \( A = \{x: x< 4, x \in W \}\) then A = {0, 1, 2, 3} and n (A) = 4
• If B = { letters in the word “mathematics”} then B = {m, a, t, h, e, i, c, s} and n(B) = 8.

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Arithmetic of Cardinality

Arithmetic of cardinality in sets refers to the mathematical operations that involve counting the number of elements (cardinality) within sets. When solving verbal problems involving sets, you might encounter situations where you need to perform arithmetic operations such as addition, subtraction, multiplication, and division on the cardinalities of sets to find the desired information.
Here are some common scenarios where arithmetic of cardinality comes into play when solving verbal problems related to sets:

1. Union of Sets: When we need to find the total number of elements in the union of two or more sets, we use the concept of cardinality. For example, if we have sets A and B, the cardinality of their union (A ∪ B) can be calculated by adding the cardinalities of A and B and then subtracting the cardinality of their intersection (A ∩ B) to avoid double counting any shared elements.
   n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
2. Complements: The complement of a set A with respect to a larger set U (universal set) contains all elements in U that are not in A. You can calculate the cardinality of the complement by subtracting the cardinality of A from the cardinality of U.
   n(A') = n(U) - n(A)
3. Subtraction of Sets: When you want to find the number of elements in one set that are not in another set, you can use subtraction of cardinalities. For instance, if you have sets A and B, the cardinality of the difference A - B is found by subtracting the cardinality of B from the cardinality of A.
   n(A - B) = n(A) - n(B)

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Test your Understandings

From the Venn-diagram given below, find the cardinal number given sets.

1. \(n(\phi) \)
   

2. \(n(A-B) \)
   

3. \(n(B-A) \)
   

4. \(n(A \cap B) \)
   

5. \(n(A \cup B)' \)
   

6. \(n(A) \)
   

7. \(n(B) \)
   

8. \(n(A') \)
   

9. \(n(B)' \)
   







10. \(n(A \triangle B) \)
    

11. \(n(A \triangle B)' \)
    

12. \(n(AUB) \)
    

13. \(n(A_0)' \)
    

14. \(n(B_0)' \)
    

15. \(n(A \cap B)' \)
    

16. \(n(U) \)
    

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MEAN

Author & Editor

Mean Nepal

November 28, 2025 Math9